In higher category theory
The statement of the Yoneda lemma has a straightforward generalization from categories to (∞,1)-categories.
For a small -category and an -functor, the composite
is equivalent to .
Preservation of limits
The -Yoneda embedding preserves all (∞,1)-limits that exist in .
This appears as HTT, prop. 18.104.22.168.
Local Yoneda embedding
For an (∞,1)-site and an (∞,1)-topos, (∞,1)-geometric morphisms from the (∞,1)-sheaf (∞,1)-topos to correspond to the local (∞,1)-functors , those that
More preseicely, the (∞,1)-functor
given by precomposition of inverse image functors by ∞-stackification and by the (∞,1)-Yoneda embedding is a full and faithful (∞,1)-functor and its essential image is spanned by these local morphisms.
This appears as (HTT, prop. 22.214.171.124).
Published statements appear in
as indicated above.
See also the discussion on MathOverflow.